Axiom 1 and 1' in LyX

Question

I'm using LyX with "Theorems (AMS, numbered by type)" and "Theorems (AMS-extended, numbered by type)" modules. I would like to have the following output:

Axiom 1. Lorem.

Axiom 1'. Ipsum.

Is there a way to do this using LyX modules + ERT? In this link (Custom enumeration for axioms or hypothesis), someone made this in pure TeX. Since I'm a begginer, I don't know how to use this to solve my problem. Maybe the solution is to edit LyX's theorem module file, but I have no idea how to do this. In this other link (How to make "claim" environments look like "theorem" environment?) there's an example of editing LyX module files to get what you want.

Answer 1

Add the following in your Local Layout of LyX (Document > Settings... > Local Layout):

#\DeclareLyXModule{Other}
#DescriptionBegin
#Defines Axiomp (primed axioms)
#DescriptionEnd
#Requires: theorems-ams-extended
#Category: theorems

Format 35

Requires    amsmath

# The environments defined (regular and starred) are :
# - Axiomp

Style Axiomp
    CopyStyle             Theorem
    DependsOn               Theorem
    LatexName             axp
    LabelString           "Axiom \theaxp."
    Preamble
      \usepackage{etoolbox}
      \theoremstyle{plain}
      \newtheorem{axp}{\protect\axiomname}
      \makeatletter
      \patchcmd{\axp}{\th@plain}{\th@plain\setcounter{axp}{\numexpr\value{thm}-1}}{}{}
      \renewcommand{\theaxp}{\arabic{axp}$'$}
      \makeatother
    EndPreamble
    LangPreamble
      \providecommand{\axiomname}{_(Axiom)}
    EndLangPreamble
    BabelPreamble
      \addto\captions$$lang{\renewcommand{\axiomname}{_(Axiom)}}
    EndBabelPreamble
End

The click validate to add Axiomp to your list of styles. While your display may resemble

your output should result in

---

If you wish to have more control over the numbering, then you can add the following to your Document > Settings > LaTeX Preamble:

\makeatletter
\renewcommand{\axp}{%
  \@@thm
    {\let\thm@swap\@gobble\th@plain}
    {axp}
    {\protect\axiomname}}

\def\@@thm#1#2#3{% modified from original \@thm
  \ifhmode\unskip\unskip\par\fi
  \normalfont
  \trivlist
  \let\thmheadnl\relax
  \let\thm@swap\@gobble
  \thm@notefont{\fontseries\mddefault\upshape}%
  \thm@headpunct{.}% add period after heading
  \thm@headsep 5\p@ plus\p@ minus\p@\relax
  \thm@space@setup
  #1% style overrides
  \@topsep \thm@preskip               % used by thm head
  \@topsepadd \thm@postskip           % used by \@endparenv
  \def\@tempa{#2}\ifx\@empty\@tempa
    \def\@tempa{\@oparg{\@@begintheorem{#3}{}}[]}%
  \else
    \setcounter{#2}{\value{ax}}%
    \def\@tempa{\@oparg{\@@begintheorem{#3}{\csname the#2\endcsname}}[]}%
  \fi
  \@tempa
}

\def\@@begintheorem#1#2[#3]{%
  \typeout{hello}%
  \def\myvar{#3}\show\myvar%
  \deferred@thm@head{\the\thm@headfont \thm@indent
    \@ifempty{#1}{\let\thmname\@gobble}{\let\thmname\@iden}%
    \@ifempty{#2}{\let\thmnumber\@gobble}{\let\thmnumber\@iden}%
    \@ifempty{#3}{}{\setcounter{axp}{#3}}%\@ifempty{#3}{\let\thmnote\@gobble}{\let\thmnote\@iden}%
    \let\thmnote\@gobble%
    \thm@swap\swappedhead\thmhead{#1}{#2}{#3}%
    \the\thm@headpunct
    \thmheadnl % possibly a newline.
    \hskip\thm@headsep
  }%
  \ignorespaces}
\makeatother

It modifies the theorem environment associated with primed axioms (axp) to reroute to a specialized \@@thm and \@@begintheorem constructions. Input of the form

yields as output

The use of an ERT for the optional argument is required, as is mentioned in the LyX wiki How to add optional arguments to environments:

Many LaTeX environments accept optional arguments that influence their behavior. The theorem environment, for example, accepts an optional argument that is typeset after the theorem number, often used for the name of the theorem or to give credit to its discoverer. In LaTeX, this would be written thus: \begin{theorem}[Cohen, 1961] or \begin{theorem}[Schröder-Bernstein Theorem], with results like these:

Theorem 1.1 (Cohen, 1961).
Theorem 1.2 (Schröder-Bernstein Theorem).

LyX offers no obvious way to do this, but in fact it is quite simple. All you need to do is enter the optional argument as ERT immediately following the appearance of "Theorem 1.2", or whatever, in LyX. It works because the optional argument follows the environment declaration.